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Programming Languages and Compilers (CS 421)

Grigore Rosu

2110 SC, UIUC

http://courses.engr.illinois.edu/cs421

Slides by Elsa Gunter, based in part on slides by Mattox Beckman, as updated by Vikram Adve and Gul Agha

Overview

This lesson we learn how to represent various programming language features within lambda-calculus Datatypes Booleans and integer numbers Recursion

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Untyped -Calculus

Only three kinds of expressions: Variables: x, y, z, w, … Abstraction: x. e (Function creation) Application: e1 e2

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How to Represent (Free) Data Structures (First Pass - Enumeration Types)

Suppose is a type with n constructors: C1,…,Cn (no arguments)

Represent each term as an abstraction:

Let Ci x1 … xn. xi

Think: you give me what to return in each case (think match statement) and I’ll return the case for the i‘ th constructor

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How to Represent Booleans

bool = True | False True x1. x2. x1 x. y. x False x1. x2. x2 x. y. y

Notation Will write x1 … xn. e for x1. … xn. e

e1 e2 … en for (…(e1 e2 )… en )

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Functions over Enumeration Types

Write a “match” function match e with C1 -> x1

| … | Cn -> xn

x1 … xn e. e x1…xn

Think: give me what to do in each case and give me a case, and I’ll apply that case

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Functions over Enumeration Types

type = C1 | … | Cn

match e with C1 -> x1

| … | Cn -> xn

match = x1 … xn e. e x1…xn

e = expression (single constructor) xi is returned if e = Ci

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match for Booleans

bool = True | False True x1 x2. x1 x y. x False x1 x2. x2 x y. y

matchbool = ?

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match for Booleans

bool = True | False True x1 x2. x1 x y. x False x1 x2. x2 x y. y

matchbool = x1 x2 e. e x1 x2

x y b. b x y

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How to Write Functions over Booleans

if b then x1 else x2 if_then_else b x1 x2 = b x1 x2

if_then_else b x1 x2 . b x1 x2

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How to Write Functions over Booleans

Alternately: if b then x1 else x2 =

match b with True -> x1 | False -> x2 matchbool x1 x2 b =

( x1 x2 b . b x1 x2 ) x1 x2 b = b x1 x2 if_then_else b x1 x2. (matchbool x1 x2 b)

= b x1 x2. ( x1 x2 b . b x1 x2 ) x1 x2 b

= b x1 x2. b x1 x2

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Example:

not b = match b with True -> False | False ->

True (matchbool) False True b

= ( x1 x2 b . b x1 x2 ) ( x y. y) ( x y. x) b

= b ( x y. y)( x y. x)

not b. b ( x y. y)( x y. x) Try and, or

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and or

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How to Represent (Free) Data Structures (Second Pass - Union Types)

Suppose is a type with n constructors: type = C1 t11 … t1k | … |Cn tn1 … tnm

Represent each term as an abstraction:

Ci ti1 … tij x1 … xn . xi ti1 … tij

Ci ti1 … tij x1 … xn . xi ti1 … tij

Think: you need to give each constructor its arguments first

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How to Represent Pairs

Pair has one constructor (comma) that takes two arguments

type (,) pair = (_ , _) (a , b) --> x . x a b (_ , _) --> a b x . x a b

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Functions over Union Types

Write a “match” function match e with C1 y1 … ym1 -> f1 y1 … ym1

| … | Cn y1 … ymn -> fn y1 … ymn

match f1 … fn e. e f1…fn

Think: give me a function for each case and give me a case, and I’ll apply that case to the appropriate function with the data in that case

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Functions over Pairs

matchpair = f p. p f

fst p = match p with (x,y) -> x fst p. matchpair ( x y. x) p

= p. ( f p. p f) ( x y. x) p = p. p ( x y. x)

snd p. p ( x y. y)

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How to Represent (Free) Data Structures (Third Pass - Recursive Types)

Suppose is a type with n constructors: type = C1 t11 … t1k | … | Cn tn1 … tnm

Suppose tih : (i.e., it is recursive) In place of a value tih , let us have a function to

compute the recursive value rih x1 … xn

Ci ti1 … rih …tij x1 … xn . xi ti1 … (rih x1 … xn) … tij

Ci ti1 … rih …tij x1 … xn . xi ti1 … (rih x1 … xn) … tij

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How to Represent Natural Numbers

nat = Suc nat | 0 Suc = n f x. f (n f x) Suc n = f x. f (n f x) 0 = f x. x Such representation called

Church Numerals

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Some Church Numerals

Suc 0 = ( n f x. f (n f x)) ( f x. x) -->

f x. f (( f x. x) f x) --> f x. f (( x. x) x) --> f x. f x

Apply a function to its argument once

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Some Church Numerals

Suc(Suc 0) = ( n f x. f (n f x)) (Suc 0) -->

( n f x. f (n f x)) ( f x. f x) --> f x. f (( f x. f x) f x)) --> f x. f (( x. f x) x)) --> f x. f (f x)Apply a function twice

In general n = f x. f ( … (f x)…) with n applications of f

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Primitive Recursive Functions

Write a “fold” function fold f1 … fn = match e with C1 y1 … ym1 -> f1 y1 … ym1

| … | Ci y1 … rij …yin -> fn y1 … (fold f1 … fn rij) …ymn | … | Cn y1 … ymn -> fn y1 … ymn

fold f1 … fn e. e f1…fn

Match in non-recursive case a degenerate version of fold

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Primitive Recursion over Nat

fold f z n= match n with 0 -> z | Suc m -> f (fold f z m)

fold f z n. n f z

is_zero n = fold ( r. False) True n = ( f x. f n x) ( r. False) True = (( r. False) n ) True if n = 0 then True else False

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Adding Church Numerals

n f x. f n x and m f x. f m x

n + m = f x. f (n+m) x = f x. f n (f m x) = f x. n f (m f x)

+ n m f x. n f (m f x)

Subtraction is harder

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Multiplying Church Numerals

n f x. f n x and m f x. f m x

n m = f x. (f n m) x = f x. (f m)n x = f x. n (m f) x

n m f x. n (m f) x

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Predecessor

let pred_aux n = match n with 0 -> (0,0) | Suc m-> (Suc(fst(pred_aux m)), fst(pred_aux m)= fold ( r. (Suc(fst r), fst r)) (0,0) n

pred n. snd (pred_aux n) n = n. snd (fold ( r.(Suc(fst r), fst r)) (0,0) n)

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Recursion

Want a -term Y such that for all terms R we have

Y R = R (Y R) Y needs to have replication to

“remember” a copy of R Y = y. ( x. y(x x)) ( x. y(x x)) Y R = ( x. R(x x)) ( x. R(x x)) = R (( x. R(x x)) ( x. R(x x))) Notice: Requires lazy evaluation

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Factorial

Let F = f n. if n = 0 then 1 else n * f (n - 1)

Y F 3 = F (Y F) 3 = if 3 = 0 then 1 else 3 * ((Y F)(3 - 1))= 3 * (Y F) 2 = 3 * (F(Y F) 2)= 3 * (if 2 = 0 then 1 else 2 * (Y F)(2 - 1))= 3 * (2 * (Y F)(1)) = 3 * (2 * (F(Y F) 1)) =… = 3 * 2 * 1 * (if 0 = 0 then 1 else 0*(Y F)(0 -

1))= 3 * 2 * 1 * 1 = 6

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Y in OCaml

# let rec y f = f (y f);;val y : ('a -> 'a) -> 'a = <fun># let mk_fact = fun f n -> if n = 0 then 1 else n * f(n-

1);;val mk_fact : (int -> int) -> int -> int =

<fun># y mk_fact;;Stack overflow during evaluation

(looping recursion?).

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Eager Eval Y in Ocaml

# let rec y f x = f (y f) x;;

val y : (('a -> 'b) -> 'a -> 'b) -> 'a -> 'b = <fun>

# y mk_fact;;- : int -> int = <fun># y mk_fact 5;;- : int = 120 Use recursion to get recursion

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Some Other Combinators

For your general exposure

I = x . x K = x. y. x K* = x. y. y

S = x. y. z. x z (y z)